How do tree diagrams work? Consider: A bag contains 5 red counters and 3 blue counters. James draws a counter from the bag at random and keeps it. James then draws a second counter at random. What is the probability that James takes two red counters?

Tree diagrams can be used to work out probabilities when you have a combination of events happening. The end of the branch displays the outcome of an event and the probability of this event occurring is written above/ below the branch. A tree diagram can have multiple branches and can be manipulated to determine the probabilities of multiple events occurring.
In the above example we want to determine the different outcomes of the first event and display this information on our tree diagram; James can either draw a red counter or a blue counter. Next we want to calculate the chances of each outcome happening. To do this we want to find the total number of counters in the bag, this would be 3 + 5 = 8 counters. As there are 5 red counters in the bag, and 8 counters in total, the chance of drawing a red counter is 5/ 8 and hence the chance of drawing a blue counter is 3/ 8. We display this information on our diagram. We then repeat this process for James' second draw. However, we must check if the events are dependent (does the outcome of the first event affect the second event in any way?) and adjust our probabilities accordingly. Once we have completed the branches of our tree, we identify the combination of the events we are interested in. In this example we are interested in the probability of James drawing a red counter followed by another red counter. To calculate this, we multiply the probabilities of the relevant branches together, in this case 5/8 x 4/7 = 20/56. We then simplify our answer to 5/14.